Physics 451, 452, 725: Mathematical Methods Russell Bloomer1 University of Virginia Note: There is no guarantee that these are correct, and they should not be copied 1 email: rbloomer@virgi...
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This booklet and more can be downloaded for free from the New Zealand Centre of Mathematics www.mathscentre.co.nz. There you can also watch over 80 videos and download over 100 similar maths revisi...
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Approximate Methods of Analysis of Multi-storey Frames
Statistical Methods
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Approximate Methods for Analysis of Indeterminate Structures (Ref: Chapter 7) During preliminary design and analysis, the actual member dimensions are not usually known. Approximate analysis is useful in determining (approximately) the forces and moments in the different members and in coming up with preliminary designs. Based on the preliminary design, a more detailed analysis can be conducted and then the design can be refined. Approximate analysis is conducted by making realistic assumptions about the behavior of the structure. Approximate Analysis of Indeterminate Trusses Case 1: Diagonals CANNOT carry compression (because they are designed to be long and slender) In this case, the truss is usually determinate because only one of the diagonals is active. The "compression" diagonal behaves as a zero-force member. Case 2: Diagonals CAN carry compression In this case, both diagonals will be assumed to carry half the panel shear.
Example Find the forces in the truss members • If diagonals cannot carry compression • If diagonals can carry compression
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Portal Frames and Trusses Case 1: Pin supported
Case 2: Fixed Supported
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Case 3: Partial Fixity
Case 4: Portal Trusses
Example Find the forces in the members of the truss
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Frame Structures with Vertical Loads
Assume hinge at 0.1 L
Example Find the moments (in the beam) at joints E and C
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Frame Structures with Lateral Loads: Portal Method For low-rise building frames under lateral loads, the frame can be viewed as a superposition of a number of portals.
Assumptions: • Internal hinges at the centers of beams and columns • Shear carried by interior columns is assumed to be twice of that of the exterior columns.
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Example Find the reactions at the base of the columns using the portal method of analysis.
Level 2
Level 1
In a similar way, proceed from the top to bottom, analyzing each of the small pieces.
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Frame Structures with Lateral Loads: Cantilever Method For tall and slender building frames under lateral loads, the entire frame acts similar to cantilever beam sticking out of the ground. Axial compression and tension forces develop to counteract the moment created due to the lateral load around the base of the building.
Assumptions: • Internal hinges form at the center of beams and columns • Axial stress in a column is proportional to its distance away from the centroid of the cross-sectional area of columns.
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Example Find the reactions at the base of the columns using the cantilever method of analysis. Note the areas of cross-sections of the columns are different.
Once κ (the constant of proportionality) is obtained, one can obtain the column forces FL, FM, FN, and FO. Then the rest of the analysis would proceed in a similar way to the Portal method, analyzing each of the small pieces between the assumed hinges from top to bottom.
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Example Using the cantilever method, find the reactions at the base of the frame. All columns have the same cross-sectional area.
Once again, each of the small pieces will have only 3 unknowns and can be solved for using Statics.
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In a similar way, proceed from the top to bottom, analyzing each of the small pieces.